Atlas › Modular forms › Weight 2 › Level 577 › 577.2.a › 577.2.a.e

Newform orbit 577.2.a.e

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q - 13q^{2} - 8q^{3} + 21q^{4} - 8q^{5} - 4q^{6} - 20q^{7} - 39q^{8} + 16q^{9} - 2q^{10} - 23q^{11} - 11q^{12} - 13q^{13} - 36q^{15} + 31q^{16} - 28q^{17} - 28q^{18} - 9q^{19} - 15q^{20} - 9q^{21} + 9q^{22} - 55q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} + 21x^{21} + 151x^{20} + 232x^{19} - 2299x^{18} - 10631x^{17} + 974x^{16} + 90983x^{15} + 121172x^{14} - 314176x^{13} - 731016x^{12} + 427619x^{11} + 1869473x^{10} + 1571x^{9} - 2435409x^{8} - 531550x^{7} + 1617912x^{6} + 445774x^{5} - 485480x^{4} - 131361x^{3} + 57400x^{2} + 12950x - 1487$

Atkin–Lehner signs

$p$	$w_p$
577	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−13	−8	−8	−20	−23	−13	−28	−9	−55	−15	−11	−14	−1	−24	−69	−46	−21	14	−29	−68	−10	−19	−42	−4	−6

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 577]: [0, 48] (certified at ell = 2147481793)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(577, 2)[4]   # 577.2.a.e

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 577:2 1000